1 Antwort

In general, if you want to compute the greatest (bi)simulation relation between two transition systems ${\cal K}_1$ and ${\cal K}_2$ over variables ${\cal V}_1$ and ${\cal V}_2$, respectively, you compute a sequence of relations ${\cal H}_0,{\cal H}_1,\ldots$ that converges towards the desired (bi)simulation relation.

To this end, you start with the greatest relation ${\cal H}_0\subseteq {\cal S}_1\times {\cal S}_2$ such that states $(s_1,s_2)\in{\cal H}_0$ have comparable labels which means that ${\cal H}_0$ contains exactly those pairs of states $(s_1,s_2)\in {\cal S}_1\times {\cal S}_2$ where ${\cal L}_1(s_1)\cap{\cal V}_2 = {\cal L}_2(s_2)\cap{\cal V}_1$ holds. This means that the labels of the states are the same when you only consider the observable variables.

If I understood your example correctly, this means (allow me to rename your state Q2 to $s_2$):

${\cal L}_2(s_2)\cap{\cal V}_1 = $ {} $\cap$ {a,b,c} = {}

So, if ${\cal L}_1(s_1)\cap$ {a,b} = {} holds (which is what you assume), then the state pair $(s_1,s_2)$ is contained in ${\cal H}_0$, otherwise, it is not contained there (and thus in none of the other ${\cal H}_i$.

If the label of S1 in K1 is {c}, and variable c is not known by states in K2 (since K2 is only defined over variables {a,b}), then state Q1 with label {} will be considered (bi)similar to S1 in the initial relation ${\cal H}_0$. So, yes, (S1,Q1) is contained in ${\cal H}_0$ under these conditions.